Exercise
$\frac{d}{dx}\left(2x+1\right)^5\left(x^4-3\right)^6$
Step-by-step Solution
1
Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=\left(2x+1\right)^5$ and $g=\left(x^4-3\right)^6$
$\frac{d}{dx}\left(\left(2x+1\right)^5\right)\left(x^4-3\right)^6+\left(2x+1\right)^5\frac{d}{dx}\left(\left(x^4-3\right)^6\right)$
Intermediate steps
2
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
$5\left(2x+1\right)^{4}\frac{d}{dx}\left(2x+1\right)\left(x^4-3\right)^6+\left(2x+1\right)^5\frac{d}{dx}\left(\left(x^4-3\right)^6\right)$
Intermediate steps
3
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
$5\left(2x+1\right)^{4}\frac{d}{dx}\left(2x+1\right)\left(x^4-3\right)^6+6\left(2x+1\right)^5\left(x^4-3\right)^{5}\frac{d}{dx}\left(x^4-3\right)$
Intermediate steps
4
The derivative of a sum of two or more functions is the sum of the derivatives of each function
$5\left(2x+1\right)^{4}\frac{d}{dx}\left(2x\right)\left(x^4-3\right)^6+6\left(2x+1\right)^5\left(x^4-3\right)^{5}\frac{d}{dx}\left(x^4-3\right)$
Intermediate steps
5
The derivative of a sum of two or more functions is the sum of the derivatives of each function
$5\left(2x+1\right)^{4}\frac{d}{dx}\left(2x\right)\left(x^4-3\right)^6+6\left(2x+1\right)^5\left(x^4-3\right)^{5}\frac{d}{dx}\left(x^4\right)$
6
Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=2$ and $g=x$
$5\left(2x+1\right)^{4}\left(\frac{d}{dx}\left(2\right)x+2\frac{d}{dx}\left(x\right)\right)\left(x^4-3\right)^6+6\left(2x+1\right)^5\left(x^4-3\right)^{5}\frac{d}{dx}\left(x^4\right)$
7
The derivative of the constant function ($2$) is equal to zero
$5\left(2x+1\right)^{4}\left(0+2\frac{d}{dx}\left(x\right)\right)\left(x^4-3\right)^6+6\left(2x+1\right)^5\left(x^4-3\right)^{5}\frac{d}{dx}\left(x^4\right)$
8
$x+0=x$, where $x$ is any expression
$5\cdot 2\left(2x+1\right)^{4}\frac{d}{dx}\left(x\right)\left(x^4-3\right)^6+6\left(2x+1\right)^5\left(x^4-3\right)^{5}\frac{d}{dx}\left(x^4\right)$
$10\left(2x+1\right)^{4}\frac{d}{dx}\left(x\right)\left(x^4-3\right)^6+6\left(2x+1\right)^5\left(x^4-3\right)^{5}\frac{d}{dx}\left(x^4\right)$
10
The derivative of the linear function is equal to $1$
$10\left(2x+1\right)^{4}\left(x^4-3\right)^6+6\left(2x+1\right)^5\left(x^4-3\right)^{5}\frac{d}{dx}\left(x^4\right)$
Intermediate steps
11
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
$10\left(2x+1\right)^{4}\left(x^4-3\right)^6+6\cdot 4\left(2x+1\right)^5\left(x^4-3\right)^{5}x^{3}$
12
Multiply $6$ times $4$
$10\left(2x+1\right)^{4}\left(x^4-3\right)^6+24\left(2x+1\right)^5\left(x^4-3\right)^{5}x^{3}$
Final answer to the exercise
$10\left(2x+1\right)^{4}\left(x^4-3\right)^6+24\left(2x+1\right)^5\left(x^4-3\right)^{5}x^{3}$
